A Topological Conjugacy of Invariant Flows on Some Class of Lie Groups. Key Words: Topological conjugacy, Invariant flows, Lie groups.

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1 Bol. Soc. Paran. Mat. (3s.) v (0000):????. c SPM ISSN on line ISSN in press SPM: doi: /bspm.v38i A Topological Conjugacy of Invariant Flows on Some Class of Lie Groups Alexandre J. Santana and Simão N. Stelmastchuk abstract: Theaimofthis paper isto classifyinvariantflowson Liegroup Gwhose Lie algebra g is associative or semisimple. Specifically, we present this classification from the hyperbolicity of the lift flows on G g. Then we apply this construction to some special cases as Gl(n,R) and affine Lie group. Key Words: Topological conjugacy, Invariant flows, Lie groups. Contents 1 Introduction 1 2 Hyperbolic condition for topological conjugacy on Lie groups 2 3 Topological conjugacy in Gl(n, R) 6 4 Invariant flows on Semisimple and Affine Lie Groups 8 1. Introduction LetGbe aliegroupwith LiealgebragandtakeA g. Considerthe dynamical system ġ = A g = R g (A), g G. (1.1) We are interested in establishing a condition to classify systems of type (1.1) via topological conjugacy, i.e., to find a homeomorphism between the state spaces that maps trajectories of one system to trajectories of the other system, preserving the parametrization by time. We restrict to the cases where g is an associative or semisimple Lie algebra. First, we study the associative case and then we apply it in the semisimple case. To develop our results we adopt the idea, due to Ayala, Colonious and Kliemann in [1], of we work with vector bundles. In the classical case, ẋ = A i x, x R n, A i gl(n,r) for i = 1,2, a fundamental hypothesis to obtain the above homeomorphism is the hyperbolicity of the systems, that is, e Ait x C i e ait x for some a i > 0, C i > 1 and for all t > 0 (see e.g. Robinson [5, ch.iv]). As in Lie groups there is not a normed space structure we lift the flows to G g, which inherits a normed vector structure from the Lie algebra g Mathematics Subject Classification: 22E20, 54H20, 37B99. Submitted May 15, Published November 08, Typeset by B S P M style. c Soc. Paran. de Mat.

2 2 A. J. Santana and S. N. Stelmastchuk In the following section, from the study of the lift flows on G g, we present a topological classification of flows on Lie group G. To be specific, we lift two systems of type (1.1) in two systems (on G g) of type ( ġ v ) = ( Rg A Av ), (1.2) whose solution is (e At g,e At v). Supposing that these two systems on G g are hyperbolic we have that they are topologically conjugate, and, as a consequence, we show that e At g and e Bt g are topologically conjugate on G (see Theorem 2.3). As application in the third section, we take G as the linear group GL(n,R), that is, we consider dynamical systems of kind Ẋ = AX where X GL(n,R) and from Theorem 2.3 we recover the classical result of dynamical system: if the real parts of generalized eigenvalues of A and B are negative, then e At and e Bt are topologically conjugate. To finish this section, we give a partial classification in case of GL(2, R). In the last section, we consider (1.1) on a semisimple Lie group and the purpose is to study the dynamical system (1.1) in the adjoint group Ad(G). In fact, we show that if the real parts of the generalized eigenvalues of ad(a) and ad(b) are negative, then e At and e Bt are topologically conjugate. From this, we can classify dynamical systems on affine groups H V, where H is a semisimple Lie group and V a n-dimensional vector space, or rather, we establish a condition to topological conjugacy of flows on H V. 2. Hyperbolic condition for topological conjugacy on Lie groups Let G be a Lie group such that its Lie algebra g is an associative algebra. This structure is natural in matrix groups. We adopt in the trivial vector bundle G g the direct product (g,v) (h,u) = (g h,v u). Our intention is to present a hyperbolic sufficient condition on g to obtain topological conjugacy between flows on G. Take the following dynamical system ġ = R g A, A g on the Lie group G. Consider, on the trivial bundle G g, the system ( ) ( ) ġ Rg A =. (2.1) v Av A direct account shows that (e At g,e At v) is a solution of (2.1) with g G and v g (see for instance [6]). Remark 2.1. Being g an associative algebra it is true that G g is a Lie group and projections of flows on G g are natural on G and g. However, for our work, the importance of associativity property is that solution of v = Av is e At v = (ta) i i=1 i! (v), which allow us to work with hyperbolic property in a easy way.

3 Topological Conjugagy 3 Before we introduce the hyperbolic property of dynamical system of form (2.1), we say what means topological conjugacy of dynamical systems in our context. Take A,B g. Two dynamical systems ġ = R g A and ġ = R g B on G are topologically conjugate if there exist a homeomorphism ψ : G G such that ψ(e At g) = e Bt ψ(g) for g G. On G g, two dynamical system of form (2.1) are topological conjugate if there is a homeomorphism Ψ : G g G g such that Ψ((e At,e At )(g,v)) = (e Bt,e Bt )Ψ(g,v) for (g,v) G g. Since hyperbolic property needs a norm, we begin by adopting a norm g in g. Then, in the trivial vector bundle G g, we consider the following norm For this reason, (g,v) = v g. (e At g,e At v) = e At v g. Hence, the behavior of the flow e At v drives the behavior of the flow (e At g,e At v) on G g. As a direct consequence, if e At v has the hyperbolic property, that is, there exists a > 0 and C > 1 such that for every t 0 we have e At v g Ce at v g, then (e At g,e At v) has the same property. Since g is a vector space, it follows that there exists a norm A on g such that e At v A e at v A (see for instance Theorem5.1 in [5, ch.iv]). We alsodenote by (, ) A the normon G g associated with A. In the following we show a technical lemma that is necessary to show our main theorem. Lemma 2.2. Let A g and assume that hyperbolic condition is satisfied, that is, there exists a > 0 such that for every t 0 we have e At v A Ce at v A. (2.2) Then there exists a unique time τ such that (e Aτ g,e Aτ v) A = 1. Proof. We first assume that the hyperbolic condition is satisfied for e At v. In consequence, the same occur with (e At g,e At v), that is, On the other hand, if t < 0 then and, consequently, (e At g,e At v) A e at (g,v) A, t 0. (2.3) e At v A e a t v A, (e At g,e At v) A e a t (g,v) A. Taking t or t we obtain (e At g,e At v) A 0 or (e At g,e At v) A, respectively. Since (, ) A is a continuous function, there is a time τ such that (e Aτ g,e Aτ v) A = 1.

4 4 A. J. Santana and S. N. Stelmastchuk We claim that this time is unique. In fact, suppose that there exist τ 1,τ 2 R such that (e Aτ1 g,e Aτ1 v) A = (e Aτ2 g,e Aτ2 v) A = 1. Assume that τ 2 τ 1. If we write τ 2 = (τ 2 τ 1 )+τ 1, then From (2.3) it follows that 1 = e Aτ2 v A = e A(τ2 τ1) e Aτ1 v A. e A(τ2 τ1) e Aτ1 v A e a(τ2 τ1) e Aτ1 v A = e a(τ2 τ2). Therefore 1 = e a(τ2 τ1) and, consequently, a(τ 2 τ 1 ) = 0. Since a > 0, we conclude that τ 2 = τ 1. In this context, we can prove the following theorem which is an adaptation of a classical result (see for instance Theorem in [2] or Theorem 7.1 in [5, ch.iv]). Theorem 2.3. Take A,B g such that e At v and e Bt v satisfy the hyperbolic property. Then (e At g,e At v) and (e Bt g,e Bt v) are topologically conjugate in G g. In consequence, e At g and e Bt g are topologically conjugate in G. Proof. We first assume that there exist a,b > 0 such that for every t 0 we have For every t < 0, it follows that (e At g,e At v) A e at (g,v) A (e Bt g,e Bt v) B e bt (g,v) B. (e At g,e At v) A e a t (g,v) A (e Bt g,e Bt v) B e b t (g,v) B. Denote by S A and S B the unit spheres in G g S A = {(g,v) : (g,v) A = 1} and S B = {(g,v) : (g,v) B = 1}. (2.4) It is clearthat the flows(e At g,e At v) and (e Bt g,e Bt v) crosss A and S B, respectively, in an unique point, by Lemma (2.2). Lett A : G g R bethe map that associatesforevery(g,v) G g the unique time t A (g,v) such that (e AtA(g,v) g,e AtA(g,v) v) A = 1. This map is well defined by Lemma (2.2). We claim that t A is continuous. In fact, take (g,v) G g and a sequence (g n,v n ) G g such that (g n,v n ) (g,v). We observe that v n v, that is, v n v A 0. For simplicity, suppose that t A (g n,v n ) > 0 for every n N. Thus we have e AtA(g,vn) (v n v) A e ata(gn,vn) v n v A 0. This gives lim n eata(gn,vn) (v n ) = lim n eata(gn,vn) v.

5 Topological Conjugagy 5 Since A is a continuous map, lim n eata(gn,vn) (v n ) A = e Alimn ta(gn,vn) v A As e AtA(gn,vn) (v n ) A = 1 we have e Alimn ta(gn,vn) v A = 1. According to uniqueness, we get lim n t A (g n,v n ) = t A (g,v). Therefore t A is a continuous map. Before we construct a homeomorphism that conjugate the flows, we show the following property: t A (e At g,e At v) = t A (g,v) t. In fact, 1 = e AtA(g,v) v A = e AtA(g,v) e A( t) e At v A = e A(tA(g,v) t) e At v A. By uniqueness, we conclude that t A (e At g,e At v) = t A (g,v) t. v We define the maps ψ 0 : S A S B by ψ o (g,v) = (g, v B ) and φ 0 : S B S A by φ o (g,v) = (g 1 v, v A ). Note that φ 0 = ψ 1 0 and that ψ 0 and φ 0 are continuous. Hence ψ 0 is a homeomorphism. Now we extend the homeomorphism ψ 0 to G g. We begin by defining the map ψ : G g G g by { (e Bt A(g,v) e ψ(g,v) = AtA(g,v) g,e BtA(g,v) ψ 0 (g,e AtA(g,v) v)) if v 0 (g,0) if v = 0. To simplify, we write ψ(g,v) = (h 1 (g),h 2 (v)) where and h 2 (v) = h 1 (g) = e BtA(g,v) e AtA(g,v) g { e Bt A(g,v) ψ 0 (g,e AtA(g,v) v) if v 0 0 if v = 0. We now proceed to show the conjugacy of the flows. In fact, ψ(e At g,e At v) = (e BtA(eAt g,e At v) e AtA(eAt g,e At v) e At g, e BtA(eAt g,e At v) ψ 0 (g,e AtA(eAt g,e At v) e At v). Using t A (e At g,e At v) = t A (g,v) t we obtain ψ(e At g,e At v) = (e B[t A(g,v) t] e A[t A(g,v) t] e At g, e B [t A(g,v) t] ψ 0 (g,e A[t A(g,v) t] e At v)) = (e Bt e Bt A(g,v) e At A(g,v) g, e Bt e B t A(g,v) ψ 0 (g,e At A(g,v) v)) = (e Bt h 1(g), e Bt h 2(v)) = (e Bt,e Bt )(h 1(g),h 2(v)) = (e Bt,e Bt )ψ(g,v). The next step is to show that ψ is continuous. We need only consider the case (g,v) with v = 0. In fact, the map ψ = (h 1,h 2 ) is continuous if v 0 because (e At g,e At v), (e Bt g,e Bt v), t A and t B are continuous. We begin by observing that

6 6 A. J. Santana and S. N. Stelmastchuk h 1 is continuous at v = 0, so we only need to show that h 2 is continuous at v = 0. Fix g G and take a sequence v n such that v n 0 in g. Our work is to show that h 2 (v n ) h 2 (0) = 0. If t A (g,v n ) is the time such that e AtA(g,vn) v n A = 1, then t n = t A (g,v n ). Let us denote u n = ψ 0 (g, Atn v n ). Thus and, for this, u n A = ψ 0 (g,e Atn v n ) A = eatn v n A e Atn v n A = 1 h 2 (v n ) A = e Btn ψ 0 (g,e Atn v n ) A e Btn A ψ 0 (g,e Atn v n ) A = e Btn A e btn 0. We conclude that h 2 (v n ) 0, hence that h 2 is continuous at 0 g, and finally that ψ is continuous. To end, we define the map { ψ 1 (e At B(g,v) e (g,v) = BtB(g,v) g,e AtB(g,v) φ 0 (g,e BtB(g,v) v)) if v 0 (g 1,0) if v = 0. It is easily verified that ψ 1 is the inverse map of ψ. It follows from the arguments above that ψ is a homeomorphism that conjugate the flows (e At g,e At v) and (e Bt g,e Bt v) in G g. It is not difficult to see that the projection of ψ on G is a homeomorphism that conjugates e At and e Bt on G. 3. Topological conjugacy in Gl(n, R) Let A be a matrix in gl(n,r) and consider the system Ẋ = AX, X Gl(n,R). This system is well posted (see for instance [6, section 2.3]). Following the idea of previous section, we consider the trivial bundle Gl(n,R) gl(n,r) and study the topological conjugacy of the flow of ( Ẋ V ) = ( AX AV that is, the flow given by (e At X,e At V), with e At = (ta) i i=1 i!. Thus, taking a norm on R n we consider the supremun norm in gl(n,r) { } Ax A = sup ;0 x Rn. (3.1) x The choice of the norm allows us to show a well-known result that relates hyperbolic property and eigenvalues of A (see e.g. Theorem 5.1 in [5, ch. IV]). ),

7 Topological Conjugagy 7 Proposition 3.1. Take A gl(n,r) and consider the equation following statements are equivalent: Ẋ = AX. The 1. There are a norm A and a constant a > 0 such that for any initial condition X Gl(n, R), the solution satisfies e At X A e at X A, for all t > for every generalized eigenvalue µ of A we have Re(µ) < 0. Proof. We first observe that every generalized eigenvalue µ of A has Re(µ) < 0 if and only if there is a norm A on R n and a constant a > 0 such that for any initial condition x R n, the solution of ẋ = Ax satisfies Thus it is sufficient to show that e A x A e at x A, for all t > 0. e At x A e at x A e At X A e at X A. In fact, suppose that e At x A e at x A whereẋ = Ax. Since e At X(x) is asolution for ẋ = Ax, it follows that e At X(x) A e at X(x) A. Thus we obtain e At X A e at X A. Conversely, suppose that e At X A e at X A. Taking x = e 1 in Definition (3.1), it follows that e At x A e at x A with x satisfying ẋ = Ax, and it assures the last assertion is hold. Now we take a norm on Gl(n,R) gl(n,r) as (X,V) A = V A. Thus, we are able to show a sufficient condition for topological conjugacy on GL(n, R). Theorem 3.2. Consider the systems Ẋ = AX and Ẏ = BY, where A,B gl(n,r). If every generalized eigenvalue of A and B has negative real part, then e At and e Bt are topologically conjugate on Gl(n,R). Proof. ByProposition3.1, wehavethatthereexistnorms A, B andconstants a,b > 0 such that e At X A e at X A e Bt Y B e bt Y B in gl(n,r). Using Theorem (2.3) we have that e At and e Bt are topologically conjugate on Gl(n, R).

8 8 A. J. Santana and S. N. Stelmastchuk As every generalized eigenvalue of A and B has negative real part it follows that Corollary 3.3. Consider Ẋ = AX and Ẏ = BY, where A,B gl(n,r), and suppose that every generalized eigenvalue of A and B has positive real part. Then e At and e Bt are topologically conjugate on Gl(n,R). Example 3.4. Take the matrix ( a b c d ) in gl(2,r). Then its eigenvalues are given by λ = (a+d)± (a d) 2 +bc 2. When (a d) 2 +bc 0 (3.2) the real part of the eigenvalues depend only on the trace a + d. Thus, supposing that two matrix A and B in gl(2,r) satisfy the inequality (3.2) we have: 1. if tr(a) < 0 and tr(b) < 0, then by previous theorem e At and e Bt are topologically conjugate; and 2. if tr(a) > 0 and tr(b) > 0, then by previous corollary e At and e Bt are topologically conjugate. 4. Invariant flows on Semisimple and Affine Lie Groups In this section, we study the topological conjugation on semisimple Lie group. Our idea is to transfer the dynamical system ġ = A g on G to one on the adjoint Lie group Ad(G) and to find a condition to topological conjugation in this group, since Ad(G) is a matrix group. We begin by assuming that G is a semisimple Lie group. We know that solution of ġ = A g is given by g(t) = e At g 0. Applying the adjoint operator in g(t) we have Ad(g(t)) = Ad(e At g 0 ) = e ad(at) Ad(g 0 ). Taking the derivative of Ad(g(t)) with respect to t we obtain Ad (g(t)) = ad(a) Ad(g(t)). Thus considering two systems ġ = A g and ḣ = B h we have the dynamical systems on Gl(g) Ad (g(t)) = ad(a)ad(g(t)) Ad (h(t)) = ad(b)ad(h(t)).

9 Topological Conjugagy 9 Suppose that every generalized eigenvalue of ad(a) and ad(b) has real part negative. From Theorem 3.2 there exists a homeomorphism ψ : Gl(g) Gl(g) such that ψ(e ad(at) Ad(g 0 )) = e ad(bt) ψ(ad(g 0 )). Since g is semisimple, it follows that Ad is an isomorphism on its image (see for instance Corollaries 5.2 and 6.2 in [3, ch.ii]). Thus we apply Ad 1 in the above equality to obtain Ad 1 (ψ(ad(e At )Ad(g 0 ))) = Ad 1 (Ad(e Bt )ψ(ad(g 0 ))) Ad 1 (ψ(ad(e At g 0 ))) = e Bt Ad 1 ψ(ad(g 0 ))). If we denote ϕ = Ad 1 ψ Ad we have Summarizing, ϕ(e At g 0 ) = e Bt ϕ(g 0 ). Theorem 4.1. Let A,B g and consider ġ = A g and ḣ = B h. Suppose that g is a semisimple Lie algebra. If every generalized eigenvalue of ad(a) and ad(b) has negative real part then e At and e Bt are topologically conjugate on G. Our final step is to work with the class of Affine groups. This idea follows from work due to Kawan, Rocio and Santana [4]. We recall the affine Lie groups and study the topological conjugacy on it. First, we need to establish a result on semisimple Lie groups. Proposition 4.2. Take A g where g is a semisimple Lie algebra. Consider the system ġ = A g. Suppose that G has a left invariant metric and denote by ρ the distance associated to this metric. If ad(a) has negative real part for every generalized eigenvalue, then there exists a positive constant a such that ρ(e At g o,e) e at ρ(g o,e). Proof. Let <,>be aleft invariantmetric on Lie group G. As Ad is an isomorphism we consider the following metric on Ad(G): < X,Y >=< d(ad 1 )X,d(Ad 1 )Y >. Let us denote by ρ 1 the distance associated to metric on Ad(G). It follows that ρ 1 (Ad(g(t)),Id) = ρ(g(t),e) for any smooth curve g(t) G. Suppose now that Ad(c(t)) satisfies the differential equation Ad (c(t)) = ad(a) Ad(c(t)). Then Proposition 3.1 assures that there are a positive constant a such that ρ 1 (Ad(g(t)),Id) = ρ 1 (e ad(a)t Ad(g 0 ),Id) e at ρ 1 (Ad(g 0 ),Id). Hence ρ(g(t),e) e at ρ(g 0,e).

10 10 A. J. Santana and S. N. Stelmastchuk Now we recall the affine Lie groups. Let V be an n-dimensional real vector space and H a Lie group that acts on V. Take the group G = H V given by the semidirect product of H and V. If Φ(t,g) = exp G (tx)g is a flow on G, where X = (A,b) g = h V and g G, we denote by Φ H the flow on H given by Φ H (t,h) = exp H (At)h with h H. Before we state and show the last result, we recall, for group actions, that a fundamental domain is a subset of the space on which the group acts such that this subset contains exactly one point of each orbit. For example, spheres S A and S B given by (2.4) are fundamental domains. Proposition 4.3. Consider the affine group G = H V with H being a semisimple Lie group with a left invariant metric. Let π : G H be the canonical projection. Take flows Φ i (t,g) = exp(tx i )g with X i = (A i,b i ), i = 1,2, on G, and suppose that every generalized eigenvalue of ad(a 1 ) and ad(a 2 ) has negative real part. Then Φ 1 and Φ 2 are topologically conjugate. Proof. We first suppose that the real part of all generalized eigenvalues of ad(a 1 ) andad(a 2 )arenegative. FromProposition4.2wehavetwounitaryspheresS 1 A 1 and S 1 A 2 in G with center g 0 and g 1, respectively, such that exp(a 1 t)g 0 and exp(a 2 t)g 1 cross these in a unique time, respectively. It means that these unitary spheres are fundamental domains to flows exp(a 1 t)g 0 and exp(a 2 t)g 1, respectively. In this way, Proposition 12 of [4] guarantees that Φ 1 and Φ 2 are topologically conjugate. References 1. Ayala, V.; Colonious, F.; Kliemann, W. On topological equivalence of linear flows with applications to bilinear control systems, J. Dynam. Control Systems, 13, (2007). 2. Colonious, F.; Kliemann, W. Dynamical System and Linear Algebra, Grad. Stud. Math., AMS, 158, (2014). 3. Helgason, S.; Differential Geometry and Symmetric Space, Academic Press, Vol. XII, (1962). 4. Kawan, C.; Rocio, O. G.; Santana, A. J. On topological conjugacy of left invariant flows on semisimple and affine Lie groups. Proyecciones, 30, (2011). 5. Robinson, C.; Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, Second edition, Stud. Adv. Math., CRC Press, Boca Raton, FL, (1999). 6. Sachkov, Y.L.; Control theory on Lie groups, J. Math. Sci., 156, (2009). Alexandre J. Santana, Department of Mathematics, State University of Maringa, Brazil. address: ajsantana@uem.br and Simão N. Stelmastchuk, Department of Mathematics, Federal University of Parana, Brazil. address: simnaos@gmail.com